ABC is a triangle . If sin `((A+B)/(2))=(sqrt(3))/(2)` then the value of sin `(C)/(2)` is

Options

A. `(1)/(sqrt(2))`
B. 0
C. `(1)/(2)`
D. `(sqrt(3))/(2)`

Answer

To solve the problem, we start with the given information about triangle ABC:

1. We know that \( \sin\left(\frac{A + B}{2}\right) = \frac{\sqrt{3}}{2} \).

2. From the sine value, we can determine the angle \( \frac{A + B}{2} \):
- Since \( \sin\left(60^\circ\right) = \frac{\sqrt{3}}{2} \), we have:
\[ \frac{A + B}{2} = 60^\circ \]

3. To find \( A + B \), we multiply both sides by 2:
\[ A + B = 120^\circ \]

4. In a triangle, the sum of the angles is \( 180^\circ \):
\[ A + B + C = 180^\circ \]
Substituting \( A + B = 120^\circ \):
\[ 120^\circ + C = 180^\circ \]

5. Solving for \( C \):
\[ C = 180^\circ - 120^\circ = 60^\circ \]

6. Now, we need to find \( \sin\left(\frac{C}{2}\right) \):
\[ \sin\left(\frac{C}{2}\right) = \sin\left(\frac{60^\circ}{2}\right) = \sin(30^\circ) \]

7. We know that \( \sin(30^\circ) = \frac{1}{2} \).

Thus, the value of \( \sin\left(\frac{C}{2}\right) \) is \( \frac{1}{2} \).

### Summary of Steps:
1. Identify the given sine value and relate it to the angle.
2. Solve for \( A + B \).
3. Use the triangle angle sum property to find \( C \).
4. Calculate \( \sin\left(\frac{C}{2}\right) \).
5. Conclude with the final value.